Sfb 450 Analysis and Control of Ultrafast Photoinduced Reactions - PDF document

Title (Notas) 13.11.02 11:12 Sfb 450 Analysis and Control of Ultrafast Photoinduced Reactions “Dissipation (dephasing and energy relaxation) in the Harmonic Oscillator” Luis Lustres AK Ernsting Instiut für Chemie Humboldt Universität zu Berlin 12 th November 2002

Lecture Organization (Notas) 13.11.02 11:12 How is this Talk Organized? Pure and Mixed States: The Statistical Operator. Coherence. Non-Perturvative Propagation of the Density Matrix ( ρ ρ ). ρ ρ A Simple Example: Propagation in the H 2 Molecule within the Harmonic Oscillator Model. Collisions: Perturbative Propagation of ρ ρ . ρ ρ The Redfield Theory of Relaxation. Linear Coupling to a Stochastic Bath Variable ( ξ ). Summary. PART 1 Difference between Pure and Mixed States The use of the Statistical or Density Operator to describe Mixed States Propagation of a Superposition State in the absence of perturbation. We illustrate this concepts with the aid of a simple example provided by the excitation of the hydrogen molecule to an ionic state with arbitrarely tunable and ultrashort laser pulses. PART 2 Dephasing, Collisions and Relaxation after creation of a superposition state Redfield theory of relaxation We illustrate the previous concepts with a simple model in which a superposition state of a low-frequency monodimensional harmonic oscillator relaxes through interaction with a stochastic bath

Pure and Mixed States (Notas) 13.11.02 11:12 Pure and Mixed States The Representation or Basis { φ i } Pure States = ∑ | Ψ〉 a i | φ i 〉 i Mixed States n | = ∑ = ∑ | Ψ〉 W n | Ψ n 〉 φ i 〉 W n a i n n,i = ∑ = ∑ ˆ ˆ ˆ ˆ 〈 Q 〉 W n 〈 Q n 〉 W n 〈 Ψ n | Q | Ψ n 〉 n n 1. Most of the methods in Quantum Mechanics based on the Variational Theorem or the Perturbation Theory use basis states. Basis states are also the esence of the Heisemberg Matrix representation of Quantum Mechanics. By means of the basis states we are able to understand how similar is our actual system to some simpler model systems whose behavour is well understood. 2. We express the wavefunction ( WF ) of our system as a linear combination of Basis States. 3. When working with ultrashort laser pulses, the excitation energy is imprecise and the state of the system after electronic optical excitation has to be described as a superposition state or wavepacket ( WP ). A WP is a linear combination of eigenstates of the system weighted by some probability. All these eigenstates can be themselves linear combinations of basis functions. 4. In the definition of the mean value of some observable Q statistics enters in two ways: in the definition of mean value and in the average over the states.

The Statistical or Density Operator (Notas) 13.11.02 11:12 The Statistical or Density Operator Properties The Density Matrix i . Hermitian ρ ii. means ensemble-averaged = ∑ ii ρ ρ ˆ ˆ W n | Ψ n 〉 〈 Ψ n | ρ ρ ˆ ˆ probability n = ∑ ρ ρ ˆ ˆ n * n | φ j 〉 〈 φ i | ρ ρ ˆ ˆ = ∑ W n a i a j Tr ( ρ ρ ρ ρ ) i ρ ii = iii. 1 n,i,j = ∑ ρ 〈 φ i | ρ ρ ˆ ˆ ˆ ˆ | φ j 〉 n a j n * ρ ρ n W n a i 〈 Q 〉 = Tr ( ρ ρ ρ ρ Q ) iv. ij ≡ 1. For the description of superposition systems it is more convenient to work with the density matrix insted of using the WF of the system. 2. The density or statistical operator is defined as the sum-over-the-states of the counter-product of the wavefunctions composing the superposition state and weighted by their respective probabilities. 3. The density matrix element ρ ij is the average value over the states of the product of the contribution coefficient of the basis state φ i by the contribution coefficient of the basis state φ j (complex conjugate). 4. Important Properties: It is hermitian. The diagonal elements ρ ii are the enesemble averaged probability of having a system described by the basis function φ i . As the meaning of the diagonal is probability the trace of the matrix has to be normalized to unity. It can be shown that the mean value of the observable Q is the trace of the matrix Q σ .

The Off-diagonal Terms-1 (Notas) 13.11.02 11:12 The Off-Diagonal Terms in a Coherent Two-level System Basis { b } φ a , φ Superposition State exp [ ħ t ] exp [ ]  E a  E b | Ψ n ( t ) 〉 = a n ( 0 ) � | φ a 〉 � b n ( 0 ) � ħ( t � n ) | φ b 〉 t 0 Density Matrix W n [ ] | a n ( 0 ) | 2 a n ( 0 ) b n ( 0 ) * exp (  Ω ba t ) = ∑ ρ ρ ρ ρ a n ( 0 ) * b n ( 0 ) exp ( �  Ω ba t ) | b n ( 0 ) | 2 n 1. Up to now we described the physical meaning of the diagonal elements of the density matrix. We now want to understand the meaning of the off-diagonal elements. 2. For simplicity, let's assume that we have a two-dimensional basis and a time-independet Hamiltonian. Any superposition state could be expressed as a linear combination of the two basis states. The time evolution of the WF is obttained by applying the time-dependent Schrödinger operator to it. 3. Now assume that we have control over the relative phases of the two basis-states. In other words, let's assume that we can “start” both states at exactly the same time for all the elements of the ensemble. 4. The values of the diagonal elements will depend only on the initial amplitudes of the basis states whereas the value of the off-diagonal elements will oscillate around the time-zero values with a frequency given by the energy difference of the basis states.

The Off-diagonal Terms-2 (Notas) 13.11.02 11:12 The Off-Diagonal Terms in a Non-Coherent Two-level System Stateto State Random Phase n E b t 0 � ħ n ≡ Density Matrix W n [ ] | a n ( 0 ) | a n ( 0 ) b n ( 0 ) * exp [ Â ( � n � Ω ba t ) ] 2 = ∑ ρ ρ ρ ρ a n ( 0 ) * b n ( 0 ) exp [ � Â ( � n � Ω ba t ) ] | b n ( 0 ) | 2 n 2 π [ ] 2 π | a n ( 0 ) | a n ( 0 ) b n ( 0 ) * exp [ Â ( � n � Ω ba t ) ] 2 „ � 1 ∫ ρ ρ = ρ ρ a n ( 0 ) * b n ( 0 ) exp [ � Â ( � n � Ω ba t ) ] | b n ( 0 ) | 2 0 1. Let´s assume now that we do not have control over the phases of the basis states or as we said before, that we can not “start” both states with some given phase difference for all the elements of the ensemble. 2. In this case, all the phase differences are equally probable and the probability of every phase in the superposition state is given by 1/2 π . 3. After integrating between 0 and 2 π we observe that the off-diagonal elements vanish and the density matrix is diagonal. 4. Non-coherent creation of a superposition state leads to a dephased state. All the process by means of which an initial coherent state losses its coherence to give an uncoherent state are call dephasing .

Quantum Beats (Notas) 13.11.02 11:12 Quantum Beats Superposition State a exp [ a t ] bexp [ b t ]  E a  E b | Ψ( t ) 〉 = � ħ t � γ | φ a 〉 � � ħ t � γ | φ b 〉 Spontaneous Fluorescence I ( t ) ∝ | 〈 0 | µ µ µ µ ˆ ˆ ˆ ˆ | Ψ( t ) 〉 | 2 = | µ | 2 0 Ψ � 2 γ � 2 γ | µ | 2 = | a µ 0a | 2 e a t � | b µ 0b | 2 e b t � 0 Ψ exp [ t ]  Ea � Eb ab * µ 0a µ 0b * � t � ( γ a �γ b ) � ħ 0b exp [ t ]  Ea � Eb a * b µ 0a * µ t � ( γ a �γ b ) ħ 1. Coherence manifests itself in the signal of our experiments but the kind of manifestation depends on the kind of measurements we do. 2. We continue with our example of the superposition of two basis states evolving under a time-independent Hamiltonian. 3. Let's consider the example of Spontaneous fluorescence . The emitted signal, in the semiclassical treatment and within the Franck-Condon approximation is proportional to the electric transition dipole moment integral of the molecule. Here, we do not take polarization into account. 4. This integral can be expanded in four terns. The two first terms would correspond to the fluorescence intensity originating from the two basis states and weighted by their respective probabilities. The last two terms oscillate with a frequency given by the energy separation of the basis states and are proportional to the off-diagonal elements of the density matrix. If the system is dephased the last two terms vanish. The appearence of the “coherences” can be understood as an interference between the basis states.